Axiom:Axiomatization of 1-Based Natural Numbers

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Axioms

The following axioms are intended to capture the behaviour of the ($1$-based) natural numbers $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations of addition $+$ and multiplication $\times$ as they pertain to $\N_{>0}$:

\((A)\)   $:$     \(\displaystyle \exists_1 1 \in \N_{> 0}:\) \(\displaystyle a \times 1 = a = 1 \times a \)             
\((B)\)   $:$     \(\displaystyle \forall a, b \in \N_{> 0}:\) \(\displaystyle a \times \paren {b + 1} = \paren {a \times b} + a \)             
\((C)\)   $:$     \(\displaystyle \forall a, b \in \N_{> 0}:\) \(\displaystyle a + \paren {b + 1} = \paren {a + b} + 1 \)             
\((D)\)   $:$     \(\displaystyle \forall a \in \N_{> 0}, a \ne 1:\) \(\displaystyle \exists_1 b \in \N_{> 0}: a = b + 1 \)             
\((E)\)   $:$     \(\displaystyle \forall a, b \in \N_{> 0}:\) \(\displaystyle \)Exactly one of these three holds:\( \)             
\(\displaystyle a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \)             
\((F)\)   $:$     \(\displaystyle \forall A \subseteq \N_{> 0}:\) \(\displaystyle \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \)             


Note

The above axiom schema specifies the old-fashioned definition of the natural numbers as:

$\text{The set of natural numbers} = \set {1, 2, 3, \ldots}$

as opposed to the more modern approach which defines them as:

$\text{The set of natural numbers} = \set {0, 1, 2, 3, \ldots}$


In order to eliminate confusion, on $\mathsf{Pr} \infty \mathsf{fWiki}$ the set $\set {1, 2, 3, \ldots}$ will be denoted as $\N_{> 0}$ or $\N_{\ne 0}$ or $\N_{\ge 1}$.

When $\N$ is used, $\N = \set {0, 1, 2, 3, \ldots}$ is to be understood.


Sources